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G = D54order 108 = 22·33

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D54, C2×D27, C54⋊C2, C9.D6, C27⋊C22, C3.D18, C6.2D9, C18.2S3, sometimes denoted D108 or Dih54 or Dih108, SmallGroup(108,4)

Series: Derived Chief Lower central Upper central

C1C27 — D54
C1C3C9C27D27 — D54
C27 — D54
C1C2

Generators and relations for D54
 G = < a,b | a54=b2=1, bab=a-1 >

27C2
27C2
27C22
9S3
9S3
9D6
3D9
3D9
3D18

Character table of D54

 class 12A2B2C369A9B9C18A18B18C27A27B27C27D27E27F27G27H27I54A54B54C54D54E54F54G54H54I
 size 11272722222222222222222222222222
ρ1111111111111111111111111111111    trivial
ρ211-1-111111111111111111111111111    linear of order 2
ρ31-11-11-1111-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41-1-111-1111-1-1-1111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ5220022222222-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-2002-2222-2-2-2-1-1-1-1-1-1-1-1-1111111111    orthogonal lifted from D6
ρ72-2002-2-1-1-1111ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989959498998998997929792979295949594    orthogonal lifted from D18
ρ8220022-1-1-1-1-1-1ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ9220022-1-1-1-1-1-1ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989    orthogonal lifted from D9
ρ10220022-1-1-1-1-1-1ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ112-2002-2-1-1-1111ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9594ζ989ζ9792989979297929792959495949594989989    orthogonal lifted from D18
ρ122-2002-2-1-1-1111ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ989ζ9792ζ9594979295949594959498998998997929792    orthogonal lifted from D18
ρ132-200-11ζ27152712ζ2724273ζ27212762715271227242732721276ζ2725272ζ2723274ζ2719278ζ272627ζ2720277ζ27142713ζ2722275ζ27172710ζ2716271127172710271627112725272272027727142713272227527232742719278272627    orthogonal faithful
ρ142200-1-1ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ2720277ζ27142713ζ272627ζ27172710ζ27162711ζ2722275ζ2723274ζ2719278ζ2725272ζ2719278ζ2725272ζ2720277ζ27162711ζ2722275ζ2723274ζ27142713ζ272627ζ27172710    orthogonal lifted from D27
ρ152-200-11ζ2724273ζ2721276ζ271527122724273272127627152712ζ2722275ζ27172710ζ2720277ζ27162711ζ2723274ζ2719278ζ272627ζ2725272ζ2714271327252722714271327222752723274271927827262727172710272027727162711    orthogonal faithful
ρ162200-1-1ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ2722275ζ27172710ζ2720277ζ27162711ζ2723274ζ2719278ζ272627ζ2725272ζ27142713ζ2725272ζ27142713ζ2722275ζ2723274ζ2719278ζ272627ζ27172710ζ2720277ζ27162711    orthogonal lifted from D27
ρ172-200-11ζ27152712ζ2724273ζ27212762715271227242732721276ζ2720277ζ27142713ζ272627ζ27172710ζ27162711ζ2722275ζ2723274ζ2719278ζ272527227192782725272272027727162711272227527232742714271327262727172710    orthogonal faithful
ρ182200-1-1ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ272627ζ2725272ζ2723274ζ27142713ζ27172710ζ2720277ζ27162711ζ2722275ζ2719278ζ2722275ζ2719278ζ272627ζ27172710ζ2720277ζ27162711ζ2725272ζ2723274ζ27142713    orthogonal lifted from D27
ρ192-200-11ζ27152712ζ2724273ζ27212762715271227242732721276ζ27162711ζ2722275ζ27172710ζ2719278ζ2725272ζ2723274ζ27142713ζ272627ζ272027727262727202772716271127252722723274271427132722275271727102719278    orthogonal faithful
ρ202-200-11ζ2721276ζ27152712ζ27242732721276271527122724273ζ272627ζ2725272ζ2723274ζ27142713ζ27172710ζ2720277ζ27162711ζ2722275ζ271927827222752719278272627271727102720277271627112725272272327427142713    orthogonal faithful
ρ212200-1-1ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ27172710ζ2720277ζ27142713ζ2722275ζ2719278ζ27162711ζ2725272ζ2723274ζ272627ζ2723274ζ272627ζ27172710ζ2719278ζ27162711ζ2725272ζ2720277ζ27142713ζ2722275    orthogonal lifted from D27
ρ222-200-11ζ2721276ζ27152712ζ27242732721276271527122724273ζ2719278ζ27162711ζ2722275ζ2723274ζ272627ζ2725272ζ2720277ζ27142713ζ2717271027142713271727102719278272627272527227202772716271127222752723274    orthogonal faithful
ρ232200-1-1ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ2723274ζ2719278ζ27162711ζ2725272ζ27142713ζ272627ζ27172710ζ2720277ζ2722275ζ2720277ζ2722275ζ2723274ζ27142713ζ272627ζ27172710ζ2719278ζ27162711ζ2725272    orthogonal lifted from D27
ρ242-200-11ζ2724273ζ2721276ζ271527122724273272127627152712ζ27142713ζ272627ζ2725272ζ2720277ζ2722275ζ27172710ζ2719278ζ27162711ζ272327427162711272327427142713272227527172710271927827262727252722720277    orthogonal faithful
ρ252-200-11ζ2721276ζ27152712ζ27242732721276271527122724273ζ27172710ζ2720277ζ27142713ζ2722275ζ2719278ζ27162711ζ2725272ζ2723274ζ27262727232742726272717271027192782716271127252722720277271427132722275    orthogonal faithful
ρ262200-1-1ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ27142713ζ272627ζ2725272ζ2720277ζ2722275ζ27172710ζ2719278ζ27162711ζ2723274ζ27162711ζ2723274ζ27142713ζ2722275ζ27172710ζ2719278ζ272627ζ2725272ζ2720277    orthogonal lifted from D27
ρ272200-1-1ζ2721276ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ2719278ζ27162711ζ2722275ζ2723274ζ272627ζ2725272ζ2720277ζ27142713ζ27172710ζ27142713ζ27172710ζ2719278ζ272627ζ2725272ζ2720277ζ27162711ζ2722275ζ2723274    orthogonal lifted from D27
ρ282-200-11ζ2724273ζ2721276ζ271527122724273272127627152712ζ2723274ζ2719278ζ27162711ζ2725272ζ27142713ζ272627ζ27172710ζ2720277ζ272227527202772722275272327427142713272627271727102719278271627112725272    orthogonal faithful
ρ292200-1-1ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ27162711ζ2722275ζ27172710ζ2719278ζ2725272ζ2723274ζ27142713ζ272627ζ2720277ζ272627ζ2720277ζ27162711ζ2725272ζ2723274ζ27142713ζ2722275ζ27172710ζ2719278    orthogonal lifted from D27
ρ302200-1-1ζ27152712ζ2724273ζ2721276ζ27152712ζ2724273ζ2721276ζ2725272ζ2723274ζ2719278ζ272627ζ2720277ζ27142713ζ2722275ζ27172710ζ27162711ζ27172710ζ27162711ζ2725272ζ2720277ζ27142713ζ2722275ζ2723274ζ2719278ζ272627    orthogonal lifted from D27

Smallest permutation representation of D54
On 54 points
Generators in S54
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)

G:=sub<Sym(54)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28)])

D54 is a maximal subgroup of   D108  C27⋊D4  Q8⋊D27
D54 is a maximal quotient of   Dic54  D108  C27⋊D4

Matrix representation of D54 in GL2(𝔽109) generated by

2251
5880
,
2251
2987
G:=sub<GL(2,GF(109))| [22,58,51,80],[22,29,51,87] >;

D54 in GAP, Magma, Sage, TeX

D_{54}
% in TeX

G:=Group("D54");
// GroupNames label

G:=SmallGroup(108,4);
// by ID

G=gap.SmallGroup(108,4);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,302,237,1203,138,1804]);
// Polycyclic

G:=Group<a,b|a^54=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D54 in TeX
Character table of D54 in TeX

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